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arXiv · 2601.07100

A dichotomy for inverse-semigroup crossed products via dynamical Cuntz semigroups

Becky Armstrong·Lisa Orloff Clark·Astrid an Huef·Diego Martínez·Ilija Tolich
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Abstract

We characterise stable finiteness and pure infiniteness of the essential crossed product of a C*-algebra by an action of an inverse semigroup. Under additional assumptions, we prove a stably finite / purely infinite dichotomy. Our main technique is the development, using an induced action, of a ``dynamical Cuntz semigroup'' that is a subquotient of the usual Cuntz semigroup. We prove that the essential crossed product is stably finite / purely infinite if and only if the dynamical Cuntz semigroup admits / does not admit a nontrivial state. Indeed, a retract of our dynamical Cuntz semigroup suffices to prove the dichotomy. Our results generalise those by Rainone on crossed products of groups acting by automorphisms of a C*-algebra, and we recover results by Kwa\'sniewski--Meyer--Prasad on C*-algebras of non-Hausdorff groupoids.

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Becky Armstrong, Lisa Orloff Clark, Astrid an Huef, Diego Martínez, Ilija Tolich. 2026-01-11. A dichotomy for inverse-semigroup crossed products via dynamical Cuntz semigroups. https://arxiv.org/abs/2601.07100

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